Inverse Fourier Transform Table

Liam Oliver Theodore

2 min read

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29-11-2024

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The Inverse Fourier Transform (IFT) is a mathematical operation that recovers a function from its Fourier transform. This table provides a concise overview of common Fourier transform pairs, showing the function and its corresponding inverse Fourier transform. Remember that these are simplified representations and may require adjustments based on specific scaling conventions and normalization factors.

Understanding the Table

Before diving into the table, it's important to understand the notation used:

• f(t): Represents the original function in the time domain.
• F(ω): Represents the Fourier transform of f(t) in the frequency domain. ω represents angular frequency (2πf, where f is frequency).
• IFT{F(ω)}: Denotes the Inverse Fourier Transform of F(ω), which returns f(t).

The table below presents several examples, focusing on common mathematical functions.

Inverse Fourier Transform Pairs

Important Considerations

• Units: Ensure consistent units throughout your calculations. The units of f(t) and F(ω) are related through the specific definition of the Fourier transform used.
• Scaling Factors: The specific form of the inverse Fourier transform depends on the chosen scaling convention (e.g., the factor of 1/(2π) or 1/√(2π) may appear in either the forward or inverse transform). Always refer to the specific definition used in your context.
• Convergence: The inverse Fourier transform exists and converges under certain conditions for the function F(ω).
• Numerical Methods: For complex functions or when analytical solutions are unavailable, numerical methods such as the Fast Fourier Transform (FFT) algorithm are used for computing the inverse Fourier transform.

This table serves as a starting point. Many other Fourier transform pairs exist, and comprehensive mathematical tables or software packages should be consulted for more extensive lists and specialized functions.